This depends on the specific Air Traffic Navigation Service (ATNS) provider providing services and the equipment the controllers are equipped with. US-based FAA controllers actually are mostly unaware of which types of surveillance data are currently contributing to the position the system is plotting aircraft on their displays or scopes.
While they have access to all elements of ADS-B data associated with an aircraft which has been received, the need to query any beyond what is included with a legacy flight strip would be the exception, not the rule. The FAA maintains the world's largest and most complex integrated air traffic surveillance system which offers multiple tiers of aircraft tracking capabilities. This network of air traffic surveillance data sources offer coverage within the vast majority of the US National Airspace System (NAS) designated as Controlled Airspace (Class E, D, C, B, & A).
The FAA does maintain a database of all US registered Mode-S 24-bit aircraft address codes, colloquially referred to as "Aircraft Hex Codes". As a function of the many Bilateral and Multilateral Aviation Safety Agreements entered into with a number of foreign nation states, the FAA also populates this database with the Mode-S Hex Code provided by these foreign Civil Aviation Authorities (CAAs) with regularly received updates.
Nearly all FAA ATC surveillance functions associated with an aircraft's Hex-Code during normal operations are automated and seamless to the controller's display or scope. These functions include the auto population of registration number, aircraft type, and last reported equipment codes, etc into the track data blocks of nonparticipating aircraft, among others.
Your question specifically asks about tracking aircraft movements using these Mode-S Hex Codes. As far as the US FAA's policy, the answer is an unquestioning YES! All air traffic track data, to include all elements of ADS-B data are recorded and retained for a range of defined duration as dictated by specific facility and role. If there is a request or need to retain any set or portion of FAA AT data beyond those disposal time-frames, this can be done indefinitely.
Every nation state CAA and ATNS provider would need to be queried individually to determine the extent of their use of collected Mode-S Hex Code data. The only specific detail I'm aware of is that the ATNS providers located in the UK (NATS) and some parts of the EU (EuroControl) generally seem to be more aware of the Hex-Codes associated with the aircraft under their control. In fact, they are known to recognize and take notice when an aircraft Hex-Code mismatch is operating within their specific sectors. An aircraft Hex-Code mismatch occurs when the aircraft's reported, filed, or communicated registration number can't be reconciled with the Mode-S Hex-Code being used and transmitted and/or the aircraft records database they maintain.
Probably more than you wanted but I'm not one for brevity!